D. V. Chudnovsky G. V. Chudnovsky H. Cohn M. B. Nathanson Editors Number Theory New York Seminar 1989-1990 With 14 Figures Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona c 1991 Springer-Verlag New York, Inc. 8 Explicit Construction of the Hilbert Class Fields of Imaginary Quadratic Fields by Integer Lattice Reduction* Erich Kaltofen1 and Noriko Yui2 Abstract Motivated by a constructive realization of generalized dihedral groups as Galois groups over Q and by Atkin's primality test, we present an explicit construction of the Hilbert class fields (ring class fields) of imaginary quadratic fields (orders). This is done by first evaluating the singular moduli of level one for an imaginary quadratic order, and then constructing the \genuine" (i.e., level one) class equation. The equation thus obtained has integer coefficients of astronomical size, and this phenomenon leads us to the construction of the \reduced" class equations, i.e., the class equations of the singular moduli of higher levels. These, for certain levels, turn out to define the same Hilbert class field (ring class field) as the level one class equation, and to have coefficients of small size (e.g., seven digits). The construction of the \reduced" class equations was carried out on MACSYMA, using a refinement of the integer lattice reduction algorithm of Lenstra-Lenstra-Lav´asz, implemented on the Symbolics 3670 at Rensselaer Polytechnic Institute. *Erich Kaltofen was partially supported by the NSF Grant No. CCR-87-05363 and by an IBM Faculty Development Award. Noriko Yui was partially supported by the NSERC Grants No. A8566 and No. A9451, and by an Initiation Grant at Queen's University. 1Department of Computer Science, Rensselaer Polytechnic Institute, Troy, NY 12180, USA. 2 Department of Mathematics, Queen's University, Kingston, Ontario, K7L3N6 Canada. AMS(MOS)(1980) Mathematics Subject Classifications (1985) Revision. Firstly:11R37, 11Y16; Secondly : 12-04, 12F10. Keywords: Hilbert class fields, Ring class fields, Class equations, Singu- lar moduli, Weber's class invariants, Generalized dihedral groups, Atkin's primality test, Integer lattice reduction algorithm. 150 E. Kaltofen and N. Yui I. INTRODUCTION I.1 Backgrounds and the main results. Singular moduli (class invariants) and class equations have been extensively studied over the years by many mathematicians. In this paper, we shall discuss the explicit construction of the class equations (the defining equations of the Hilbert class fields (the ring class fields)) of imaginary quadratic fields (orders) over Q. The \genuine" class equations having the singular moduli of the elliptic modular j-invariant as roots|the class equations of level one|have integer coefficients of astronomical size, although their constant terms and discriminants are highly divisible numbers with small prime factors (Duering [D1] and Gross-Zagier [G-Z1]; see Theorem (A2.1) below). The main theme of this paper is to present an algorithm for the construction of the \reduced" class equations which have very small coefficients and define the same Hilbert class fields (ring class fields) over Q as the \genuine" ones, using a refinement of the integer lattice reduction algorithm (the L3-algorithm), implemented with MACSYMA on the Symbolics 3670. Let τ be a imaginary quadratic number which is a root of the quadratic equation az2 + bz + c = 0 with a; b; c Z. We assume that Im τ > 0. We define the discriminant of τ to be d = disc(τ) = b2 = 4ac < 0. Let 2 K = Q(τ) = Q(pd). Let = Z b+pd be an imaginary quadratic order of K of class number h(d) =: h. O 2 Let j(z) be the elliptic modular j-inh variani t. Then the singular modulus j(τ) for τ is an algebraic integer 2 O of degree h over Q, called a class invariant of . The minimal polynomial, H , of j(τ) is know as the class O d equation of K = Q(pd), which we shall call the \genuine" class equation or the class equation of level O ⊂ one. The splitting field of Hd over Q is the field K(j(τ)) which is the ring class field of conductor f over 2 K = Q(pd), where d = dK f with dK a fundamental discriminant of K. By the Artin reciprocity theorem, the Galois group Gal(H =Q) is isomorphic to the generalized dihedral group Pic( ) o C where Pic( ) d O 2 O denotes the ideal class group of . O Weber [W] considered the explicit construction of the field K(j(τ)) = Q(τ; j(τ)); τ using other 2 O modular functions (of higher level) f(z). When f(τ) does lie in K(j(τ)) = Q(τ; j(τ)); f(τ) is also called a class invariant of , and its minimal polynomial, h , over Q, is called the \reduced" class equation or the O d class equation of higher level. The polynomial hd has small integer coefficients and (from its construction) defines the same ring class field as Hd over Q. Weber [W] initiated the construction of the reduced class equations of K = Q(pd); d < 0 with O ⊂ d 1 or 5 (mod 8) and carried out computations for 65 values of d; the largest class number treated by ≡ Weber was 7. Berwick [B] computed the singular moduli of degree 2 and 3. However, it was Watson who Explicit Construction of the Hilbert Class Fields 151 first discussed systematically the explicit construction of class invariants and reduced class equations, in his series of papers [W1, W2, W3, W4]. Watson's algorithm was based on finding all the roots of the class equations that were to be constructed. For instance, for d 1(mod 8) (with d prime), the Watson class ≡ j j equation of degree h had the single real root, f(pd)=p2, where f(z) was a Weber function (see B1.1)), and Watson described the complex roots only up to cubic conjugation. Thus, Watson had to choose from three candidates for each of the h 1 complex roots. The \right" choice was tested for whether the h 1 complex − − roots and the single real root added up to approximately a rational integer. However, this trial and error h 1 method required 3 − test sums. And, in fact, Watson stopped his construction at h = 19. There are other methods for constructing class equations of imaginary quadratic orders. One of the meth- ods is based on function theoretic arguments and utilizes the Schl¨afli modular equations. This approach was used by Schl¨afli, Weber [W] ( 73-75), Hanna [H], and by others. Hanna [H] tabulated all of the known Schl¨afli x modular equations, and constructed class equations of imaginary quadratic orders K = Q(pd); d < 0, O ⊂ with discriminants d > 239. Construction of the class fields by arithmetic means was also carried out by − Herz for unramified cyclic extensions of small degree, e.g., 4, 8, ... , in [B-C-H-I-S]. Recently, Cox [Cx] has written a book in which he discusses, among other things, computation of the class equations of level one, in the framework of primes of the form x2 + ny2 (n > 0) (see (I.3) below). In this paper, we take up the task of explicit construction of the \reduced" class equations of imaginary quadratic orders K = Q(pd); d = d f 2 < 0 with 3 - d, from where Watson left off, by presenting an O ⊂ K algorithm based on integer lattice reductions. The paper consists of two parts, Part A and Part B. Part A exposes the construction of the \genuine" (level one) class equations Hd by three different methods. The construction of the polynomials H is illustrated for the maximal order K = Q(p 719) with h = 31. d OK ⊂ − Part B is concerned with the construction of the \reduced" class equations of imaginary quadratic orders K = Q(pd). Geometrical aspects of the \reduced" class equations may be explained as follows. The O ⊂ elliptic modular function j(z) gives a complex analytic isomorphism (the uniformizer) between the compact Riemann surfaces 1 j : H∗=Γ P (C); z j(z); −! −! of genus zero, where H = z C Im z > 0 ; f 2 j g a b Γ = P SL (Z) = a; b; c; d Z; ad bc = 1 = I 2 c d j 2 − 2 and 1 H∗=Γ = H=Γ P (Q) : [ 152 E. Kaltofen and N. Yui Take a suitable subgroup G of Γ of finite index such that the associated compact Riemann surface H∗=G = a b H=G cusps of G is again of genus zero (e.g., G = Γ (2i) = c 0 (mod 2i) for i = 0; 1 4). [ f g 0 c d j ≡ − Then there is again a complex analytic isomorphism 1 u : H∗=G P (C) (u = j) G −! Γ which we shall call the uniformizer of higher level. Now let K = Q(pd); d < 0; d 5 (mod 8) and 3 - d O ⊂ 6≡ be an imaginary quadratic order of class number h. Then the singular modulus u (τ) for τ where G 2 O \ FG is a fundamental domain for G, is an algebraic integer of degree h over Q, and the minimal polynomial FG of u (τ) defines the field K(u (τ)). If u (τ) K(j(τ)), then K(u (τ)) is isomorphic to the ring class field G G G 2 G K(j(τ)) of . This gives rise to the \reduced" class equation for . The \reduced" class equation is not O O always equivalent to the \genuine class" equation under Tschirnhausen transformation. Note, however, that if K = Qpd; d < 0, has odd prime class number h, then the reduced class equation is Tschirnhausen O ⊂ equivalent to the genuine one (cf.